LMFDB - Newform orbit 700.4.a.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [700,4,Mod(1,700)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("700.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(700, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 700 = 2^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	700.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-1,0,0,0,7,0,-26,0,-37] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$41.3013370040$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$

$=$

$ q - q^{3} + 7 q^{7} - 26 q^{9} - 37 q^{11} + 38 q^{13} - 35 q^{17} + 73 q^{19} - 7 q^{21} - 64 q^{23} + 53 q^{27} + 226 q^{29} + 108 q^{31} + 37 q^{33} - 360 q^{37} - 38 q^{39} + 279 q^{41} - 32 q^{43}+ \cdots + 962 q^{99}+O(q^{100}) $

q - q^3 + 7 * q^7 - 26 * q^9 - 37 * q^11 + 38 * q^13 - 35 * q^17 + 73 * q^19 - 7 * q^21 - 64 * q^23 + 53 * q^27 + 226 * q^29 + 108 * q^31 + 37 * q^33 - 360 * q^37 - 38 * q^39 + 279 * q^41 - 32 * q^43 + 222 * q^47 + 49 * q^49 + 35 * q^51 + 508 * q^53 - 73 * q^57 + 420 * q^59 - 610 * q^61 - 182 * q^63 - 825 * q^67 + 64 * q^69 + 190 * q^71 + 275 * q^73 - 259 * q^77 + 742 * q^79 + 649 * q^81 + 1041 * q^83 - 226 * q^87 + 1417 * q^89 + 266 * q^91 - 108 * q^93 + 106 * q^97 + 962 * q^99

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

1.1

−1.00000

7.00000

−26.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$5$	$ +1 $
$7$	$ -1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	700.4.a.f	✓	1
5.b	even	2	1		700.4.a.h	yes	1
5.c	odd	4	2		700.4.e.h		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
700.4.a.f	✓	1	1.a	even	1	1	trivial
700.4.a.h	yes	1	5.b	even	2	1
700.4.e.h		2	5.c	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(700))$:

$ T_{3} + 1 $

T3 + 1

$ T_{11} + 37 $

T11 + 37

$ T_{13} - 38 $

T13 - 38

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 1 $ T + 1
$5$	$ T $ T
$7$	$ T - 7 $ T - 7
$11$	$ T + 37 $ T + 37
$13$	$ T - 38 $ T - 38
$17$	$ T + 35 $ T + 35
$19$	$ T - 73 $ T - 73
$23$	$ T + 64 $ T + 64
$29$	$ T - 226 $ T - 226
$31$	$ T - 108 $ T - 108
$37$	$ T + 360 $ T + 360
$41$	$ T - 279 $ T - 279
$43$	$ T + 32 $ T + 32
$47$	$ T - 222 $ T - 222
$53$	$ T - 508 $ T - 508
$59$	$ T - 420 $ T - 420
$61$	$ T + 610 $ T + 610
$67$	$ T + 825 $ T + 825
$71$	$ T - 190 $ T - 190
$73$	$ T - 275 $ T - 275
$79$	$ T - 742 $ T - 742
$83$	$ T - 1041 $ T - 1041
$89$	$ T - 1417 $ T - 1417
$97$	$ T - 106 $ T - 106
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Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	700.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
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